INSTITUTE OF MATHEMATICAL GEOGRAPHY
MONOGRAPH SERIES
VOLUME 20
SOLSTICE VI:
AN ELECTRONIC JOURNAL
OF
Ann Arbor, MI
1995
ISBN: 1-877751-58-8
Founding Editor-in-Chief: Sandra Lach Arlinghaus, University of Michigan; and, Institute of Mathematical Geography (independent) Editorial Advisory Board: Geography. Michael F. Goodchild, University of California, Santa Barbara Daniel A. Griffith, Syracuse University Jonathan D. Mayer, University of Washington (also School of Medicine) John D. Nystuen, University of Michigan Mathematics. William C. Arlinghaus, Lawrence Technological University Neal Brand, University of North Texas Kenneth H. Rosen, A. T. & T. Bell Laboratories Engineering Applications. William D. Drake, University of Michigan Education. Frederick L. Goodman, University of Michigan Business. Robert F. Austin, Austin Communications Education Services. Technical Editor: Richard Wallace, University of Michigan. Web Consultant: William E. Arlinghaus
WebSite: http://www-personal.umich.edu/~sarhaus/image
Electronic address: sarhaus@umich.edu
MISSION STATEMENT
The purpose of Solstice is to promote interaction between geography and mathematics. Articles in which elements of one discipline are used to shed light on the other are particularly sought. Also welcome are original contributions that are purely geographical or purely mathematical. These may be prefaced (by editor or author) with commentary suggesting directions that might lead toward the desired interactions. Individuals wishing to submit articles or other material should cont act an editor, or send e-mail directly to sarhaus@umich.edu.
SOLSTICE ARCHIVES
Back issues of Solstice are available on the WebSite of the Institute of Mathematical Geography, http://www-personal.umich.edu/~sarhaus/image and on the GOPHER of the Arizona State University Department of Mathematics. Thanks to B ruce Long for taking the initiative in this matter. The connections to this GOPHER are available along a variety of routes through the Internet.
PUBLICATION INFORMATION
The electronic files are issued yearly as copyrighted hardcopy in the Monograph Series of the Institute of Mathematical Geography. This material will appear in Volume 20 in that series, ISBN to be announced. To order hardcopy, and to obtain current price lists, write to the Editor-in-Chief of S olstice at 2790 Briarcliff, Ann Arbor, MI 48105, or call 313-761-1231.
Suggested form for citation: cite the hardcopy. To cite the electronic copy, note the exact time of transmission from Ann Arbor, and cite all the transmission matter as facts of publication. Any copy that does not superimpose pr ecisely upon the original as transmitted from Ann Arbor should be presumed to be an altered, bogus copy of Solstice.
TABLE OF CONTENTS 1. FIFTH ANNIVERSARY OF SOLSTICE 2. NEW FORMAT FOR SOLSTICE AND NEW TECHNICAL EDITOR. 3. MOTOR VEHICLE TRANSPORT AND GLOBAL CLIMATE CHANGE: POLICY SCENARIOS RICHARD WALLACE 4. EXPOSITORY ARTICLE. DISCRETE MATHEMATICS AND COUNTING DERANGEMENTS IN BLIND WINE TASTINGS JOHN D. NYSTUEN, SANDRA L. ARLINGHAUS, WILLIAM C. ARLINGHAUS
2. NEW FORMAT FOR SOLSTICE AND NEW TECHNICAL EDITOR With this issue, we work to make Solstice available to a wider readership. For the first five years, all articles were typeset using TeX, the typesetting program of Donald Knuth and the American Mathematical Society. Our goal is to continue to provide text that is available to a wide variety of readers; thus, we do transmit directly so that those without Gopher access can read Solstice. Surely one great advantage of e-mail is the ease with which it can deliver information to points remote from its source. We also wish to push the text delivery in the directions of current technology, as well. Richard Wallace has kindly agreed to serve as Technical Editor of Solstice, working in conjunction with the Editor-in-Chief, to continue to develop innovative presentations that take advantage of current technology. With this issue, we transmit a separate packet of figures to accompany the single text file. In the future, we hope to have World Wide Web access to Solstice, in addtion to the direct delivery via e-mail and continuing archiving on a Gopher. When the mathematics used requires it, we intend to offer that notation within the direct e-mail transmission (as we have in the past).
4.
EXPOSITORY ARTICLE DISCRETE MATHEMATICS AND COUNTING DERANGEMENTS IN BLIND WINE TASTINGS
JOHN D. NYSTUEN
College of Architecture and Urban Planning
The University of Michigan
SANDRA L. ARLINGHAUS
School of Natural Resources and Environment
The University of Michigan
WILLIAM C. ARLINGHAUS
Department of Mathematics and Computer Science
Lawrence Technological University
The statistician Fisher explained the mathematical basis for the
field of "Design of Experiments" in an elegant essay couched in the
context of the mathematics of a Lady tasting tea (Fisher, in Newman 1956;
Fisher 1971). In Fisher's text, the problem is to analyze completely the
likelihood that the Lady can determine whether milk was added to the tea
or tea added to milk. Problems associated with the tasting of wines have
a number of obvious similarities to Fisher's tea-tasting scenario. We
offer an analysis of this related problem, set in the context of Nystuen's
wine tasting club. To begin, a brief background of the rules of that club
seems in order; indeed, it is often the case that the application is
forced to fit the mathematics in order to illustrate the abstract. Here,
it is the real-world context that guides the mathematics selected.
Wine Tasting Strategy
The Grand Crew wine club of Ann Arbor has been blind-tasting wines
monthly for years. In a blind tasting, several wines are offered with
their identity hidden. Not only are labels covered, but the entire bottle
is covered as well because the shape and color of the bottle provides some
clues as to the identity of the wine. The wines are labeled 1 through n
in the order presented. Six to eight wines are tasted at a sitting.
Members sip the wines and score each on a scale from 1 to 20, using a
scoring method suggested by the American Wine Association. The wines are
judged on the basis of quality and individual taster preference. The
evening's host is in charge of choosing and presenting the wines. Usually
wines of a single variety but from different vineyards, wineries, prices,
or distributors are tasted. Two sheets of paper are provided to each
taster. One is a blank table with a row for each wine numbered 1 through
n in the order presented. The columns on this sheet provide space for
comments, the individual's numerical ratings of the wines, average ratings
of the group, and the range in scores for each wine. One column is
reserved for the member's guess as to the identity of the wine. The
second sheet contains information about each wine to be used to match the
wines tasted. On this sheet the wines are labeled a, b, c, and so forth,
along with information on age, winery, negotiant, and price. The tasters
try to match the identity of the wine with their individual rating on
sheet 1.
The wines are listed in unknown order on the second sheet. The
tasters make their decisions by matching the letter identification with
the numerical order of presentation. On rare occasions one or more
members correctly identifies every wine. More often two or more wines are
mislabeled, and quite often the identities seem hopelessly scrambled.
Guessing at random would seem just as effective. The question then
arises, "what are the chances of getting one, two, more, or all correct by
chance alone?" Discrete mathematics and the algebra of derangements
provides the answer to this question.
Probabilities are a matter of counting. In what proportion does a
particular combination of correct and incorrect identifications occur
purely at random out of all possible combinations? The denominator in
this proportion is a count of all possible arrangements and the numerator
is a count of all possible ways a particular event occurs, such as one
right, all the rest wrong. The denominator is easily determined. If one
has five things any of the five might be chosen first; there remain four
things any of which might be chosen next. The process continues until the
last stage in which only one can be chosen. Thus, there are 5*4*3*2*1=120
ways to arrange five bottles of wine--the customary notation for this
product is 5! (read five factorial). This notation extends to arbitrarily
large positive integers in the obvious way; 0! is defined to be 1. The
factorial of a number grows rapidly with an increase in the size of the
number; thus, 7!=5040 while 8!=40,320.
The numerator of the proportion sought is not found as easily.
Consider the case of a blind tasting of three bottles of wine. Suppose
the first one is correctly identified; the remaining two outcomes must be
both right or both wrong. It is not possible to identify two wines
correctly and the third one incorrectly. Table 1 illustrates all possible
patterns of identification for three bottles of wine, a, b, and c, with
bottle "a" presented first, bottle "b" presented second, and bottle "c"
presented third. As this table indicates, there is only one arrangement
in which all are correct, two arrangements with none correct and three
arrangements with two correct. There are, of course, no arrangements with
exactly one correct.
Table 1. All possible arrangements of three items, a, b, and c.
Number of matches and non-matches to the arrangement abc.
Matches Non-matches
abc 3 0
acb 1 2
bac 1 2
bca 0 3
cab 0 3
cba 1 2
When all possible outcomes, shown in Table 1, are enumerated, it is
an easy matter to calculate the probability of each type of event-- to
obtain the probability, divide each outcome from Table 1 by 3!, the number
of total possible arrangements. Table 2 shows the probability of each
outcome: P(0) denotes none right, P(1) denotes exactly one right, and so
forth. The sum of all probabilities adds to 1.00, as it should.
Table 2. Probability of a correct labeling.
P(0) = 2/6 = 0.33
P(1) = 3/6 = 0.50
P(2) = 0/6 = 0.00
P(3) = 1/6 = 0.17
A total enumeration approach to finding the probabilities is
satisfactory for introductory purposes and for very small samples. Even
for six, seven, or eight wines at a single tasting it is, however, not
satisfactory; Table 1 would expand to 720, 5040, or 40,320 columns for
each of those cases. Clearly more clever and mathematically elegant ways
of counting, rather than brute force listings, are required. In this
latter regard, one is reminded of the story of Gauss who, as a young
child, astounded his German schoolteacher with an instant result for what
the teacher had planned as a tedious exercise. The teacher, in order to
keep his students busy, told them to add all the numbers from 1 to 100.
Gauss immediately wrote the answer on his slate. He had apparently
discovered for himself that the sum, S, of the first n positive integers
is given by the recursive relationship S=(n(n+1)/2). Thus, all he had to
do was multiply 50 by 101 to obtain the answer: an elegant solution to an
otherwise tedious problem. It was the more mature Gauss and later Laplace
that would do pioneering work in the Theory of Errors of Observation which
in turn would serve as a significant part of the base for applications of
mathematics and statistics (in Design of Experiments) in the Scientific
Method.
Derangements
For our problem, we need a way to count the number of times a taster
can get all the wines right, one wine right and all the others wrong, two
wines right and all the others wrong, and so forth. To convert the
tedious, brute force task of listing permutations and combinations for
this problem, to a more tractable situation, we employ the concept of
"derangement," that will eliminate, notationally, combinations that we do
not wish to consider.
A "derangement" is a permutation of objects that leaves no object in
its original position (Rosen 1986; Michaels and Rosen 1991). The
permutation badec is a derangement of abcde because no letter is left in
its original position. However, baedc is not a derangement of abcde
because this permutation leaves d fixed. Thus, the number of times a wine
taster gets all the wrong answers in tasting n bottles is the number of
derangements of n numbers, D(n), divided by n!: D(n)/n!. The value of
D(n) is calculated as a product of n! and a series of terms of alternating
plus and minus signs:
D(n)=n!(1-1/1!+1/2!-1/3!+1/4!-1/5!+...+((-1)^n)/n!).
Readers wishing more detail concerning this formula might refer to Rosen
(1988); for the present, we continue to consider the use of derangements.
In order to see how derangements can be enumerated visually, we
construct the following tree of possibilities for arrangements of 5
letters which do not match the natural order of abcde. On the first
level, the natural choice is a--so choose some other letter instead. The
second level would be b in the natural order so choose all others,
instead, and continue the process until all possibilities have been
exhausted. Following each path through the tree will give all possible
derangements beginning with the letter b--there are 11 such routes. Thus,
there are 11*4 derangements.
Tree of derangements for 5 bottles.
Indeed, when there are five wines,
D(n)=5!(1/2!-1/3!+1/4!-1/5!)=5!/2!-5!/3!+5!/4!-5!/5!=60-20+5-1=44.
What is of particular significance is that derangements focus only on
wrong guesses: because a non-wrong guess is a correct guess, it is
possible to focus only on one world. The Law of the Excluded Middle, in
which any statement is "true" or "false"--with no middle partial truth
admitted, is the basis for this and for most mathematical assessments of
real-world situations. It is therefore important to use the tools
appropriately, on segments of the real-world situation in which one can
discern "black" from "white."
Derangements and Probability in Random Guesses
In the case of the five wine example, the number of ways of choosing
(for example) three correctly out of five is the combination of five
things taken three at a time: C(5,3)=5!/2!3!=10. Exhausting all possible
combinations reflects an expected connection with the binomial
theorem--these values are the coefficients of (x+y)^5.
C(5,0)=1
C(5,1)=5
C(5,2)=10
C(5,3)=10
C(5,4)=5
C(5,5)=1.
The total number of right/wrong combinations is therefore 2^5 or 32.
Notice, though, that the pattern within each grouping is disregarded; to
discover the finer pattern, of how right/wrong guesses are arranged we
need permutations. To limit the number of permutations necessary to
consider, we investigate the derangements.
If we can count derangements, we can now address the question of how
many times a taster, guessing randomly, gets exactly one wine correct. The
answer is simply the number of ways one bottle can be chosen from n
bottles times the number of derangements of the other (n-1) bottles of
wine. When this value is divided by n!, the probability P(1) of guessing
exactly one wine correctly is the result. That probability is:
P(1)=(n!/1!(n-1)!)*D(n-1)/n!
This idea generalizes in a natural manner so that the probability of
choosing exactly k wines correctly is given as:
P(k)=(n!/k!(n-k)!)*D(n-k)/n!
Table 3 displays all the probabilities for outcomes in blind tastings in
which random choices are made in situations for which from 2 to 8 wines
are offered by the evening's host. Notice that there is less than a one
percent chance of guessing all wines correctly by chance alone whenever
the host offers five or more wines in the evening's selection. Evidently,
some knowledge of wines is displayed by a taster who accomplishes this
feat with any regularity. On the other hand, one could expect, by chance
alone, to guess none of the wines correctly about 37 percent of the time,
independent of the number of wines offered for tasting. The same situation
holds for guessing exactly one wine correctly.
The reason that this is so, as readers familiar with infinite series
will note, is that the alternating series contained in the parenthetical
expression in the formula for counting derangements is precisely 1/e,
where e is the base of natural logarithms (a transcendental number of
value approximately 2.71828). That is, e^x = 1+x/1!+(x^2)/2!+(x^3)/3!+...
so that when x=-1, then e^(-1), or 1/e, is precisely the parenthetical
expression in the formula for D(n). The larger the value of n, the closer
the approximation to 1/e=0.3678797. In a blind tasting with an infinite
number of bottles of wine, random choices will result in approximately a
0.368 probability that all will be in error!
Table 3. Probability of correctly matching K wines from tasting a total
of n wines
Table 3 suggests some rules of thumb about how well a taster has done. In
a normal-sized tasting of six, seven, or eight wines, identifying at least
five of them correctly occurs less than 1% by chance alone. Identifying
four correctly happens by chance about 2 percent (or less) of the time.
However, identifying three correctly occurs by chance from near five to
six percent of the time: in every 16 to 18 tastings. Usually there are
ten to twelve tasters at a sitting in this one club. None to one member at
a sitting rates to guess three wines correctly by chance alone; the group
usually does substantially better than this, suggesting some expertise in
identifying the wines.
The Principle of Inclusion and Exclusion: The Basis for Counting.
The expression for counting derangements, as a product of n! and and a
truncated series for 1/e, has some interesting properties, most notably
perhaps the alternating plus and minus signs preceding terms of the
series. This alternation occurs because the principle of inclusion and
exclusion has been used as the basis for the counting.
Readers versed in elementary set theory, Boolean algebra, or symbolic
logic, are familiar with the idea of including the intersection, and then
subtracting it out, in order to count the number of elements in
intersecting sets. This idea, in this context, was clearly familiar to
Augustus DeMorgan in the late nineteenth century. Indeed, in a wider
context, it dates back to the time of Eratosthenes of Alexandria and his
sieve for determining which numbers are prime: those that are multiples
of numbers early in the ordering of positive integers are excluded. Only
those numbers not excluded have divisors of only themselves and 1, and so
are exactly the set included as prime numbers.
The following example illustrates how inclusion and exclusion is used
in counting derangements; the reader interested in the general proof is
referred to Rosen (1988). It is easy to visualize cases when n is small
using Venn diagrams--thus, the linkage between inclusion/exclusion, set
theory, and derangements becomes clear.
Consider for example, a tasting of two wines. Let a be the event that
the first wine is correctly identified; let b be the event that the second
wine is correctly identified. Draw a rectangle on a sheet of paper and
within the rectangle draw two intersecting circles, a and b--a familiar
Venn diagram. The content of the rectangle is the universe of discourse.
The content of circle a is the set of all events that the first wine is
correctly identified (either alone or with another), denoted N(a). The
content of circle b is the set of all events that the second wine is
correctly identified, denoted N(b). The intersection of the two circles
has content ab, the set of all events in which both the first wine and the
second wine are correctly identified, denoted N(ab). The set of all
derangements is the content of that area of the rectangle outside the two
circles. The content of the two circles is the sum of the content of the
first circle plus the sum of the content of the second circle: N(a)+N(b).
This sum however includes N(ab) in the first term and also N(ab) in the
second term; thus, N(ab) must be excluded from the sum to get an accurate
count of the content of the union of the two circles--hence inclusion and
exclusion. The accurate count of one or more wines correct is thus given
as N(a)+N(b)-N(ab). The case for three circles is more complicated to
visualize but can be enumerated carefully as a set of three two-circle
problems. With values greater than 3, visualization in this manner
becomes impossible and one must rely on extension of the notation and
visualization in the world of language rather than in the world of
pictures--both subsets of "the world of mathematics." Indeed, geographers
interested in spatial statistics should be familiar with this issue in
using the statistical forms to capture what becomes increasingly
toocomplex to map.
Retrospect
These classical ideas, whether cast in the number theoretic context of
prime numbers, in the discrete mathematics context of inclusion and
exclusion, or in the set theoretic context of intersections, served once
again, when cast in the context of derangements and the counting of
incorrectness, to permit a clever solution to a complicated, uncontrived,
real-world problem. What this sort of analysis offers is a challenge to
look at the world in different ways: from the use of classical
theoretical material in new real world situations, to the development of
new theoretical material which can foster further theoretical exploration
and application.
References
Fisher, R. A. The Design of Experiments. Eighth edition, reprinted,
New York, Hafner and Co., 1971. First edition, Edinburgh, London, Oliver
and Boyd, 1935.
Fisher, R. A. "The Mathematics of a Lady Tasting Tea" in Newman, J.R.,
ed. The World of Mathematics, Simon and Shuster, New York, 1956 (pp.
1512-1521).
Michaels, John G. and Rosen, Kenneth H. (eds.) Applications of Discrete
Mathematics, New York, McGraw-Hill, 1991.
Polya, G.; Tarjan, R. E.; and, Woods, D.R. Notes on Introductory
Combinatorics, Boston, Birkhauser, 1983.
Rosen, Kenneth R. Discrete Mathematics and Its Applications. First
edition, New York, Random House, 1988.
5.INDEX SOLSTICE: AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS WINTER, 1995 VOLUME VI, NUMBER 2 ANN ARBOR, MICHIGAN TABLE OF CONTENTS 1. MISSION STATEMENT AND BEST E-ADDRESS 2. SOLSTICE ARCHIVES 3. PUBLICATION INFORMATION 4. ELEMENTS OF SPATIAL PLANNING: THEORY. PART I. SANDRA L. ARLINGHAUS 5. MAPBANK: AN ATLAS OF ON-LINE BASE MAPS SANDRA L. ARLINGHAUS WITH ZIPPED SAMPLE ATTACHED 6. INTERNATIONAL SOCIETY OF SPATIAL SCIENCES 7. INDEX TO VOLUMES I (1990) TO V (1994); VOL. VI, NO. 1.
4. ELEMENTS OF SPATIAL PLANNING: THEORY. PART I.**
SANDRA L. ARLINGHAUS
One reason that planning of any sort is a difficult process is that
it involves altering natural boundaries to fit human needs and desires.
While it may not be "nice to fool Mother Nature" the act of planning may
be predicated on such an attempt, especially when the balance between
human and environmental needs is tipped strongly toward the human side.
At a very general level, planning how to use the Earth's surface involves
what space to use and when to use it. The "what" issues are those that
involve spatial planning; they typically involve the concept of scale. The
"when" issues involve temporal planning; they typically involve the
concept of sequence.
Particular spatial issues might address whether or not boundaries of
a parcel of land are clearly designated with respect to one's neighbors;
whether or not a proposed land use is consistent with the general
character of a larger region; or whether or not a developer's site plans
give sufficient attention to natural features. Temporal issues might
address the long range and the short range view of a traffic circulation
pattern; the sequence, in years, in which lands are to be annexed to a
city; or the length of time trees need to have lived in order to be
designated landmark trees. When one considers that budget concerns often
function as an underlying factor that can help to sway this balance, the
fragility of the art of planning becomes apparent.
One way to view complicated issues is to consider them at an abstract
level in order to understand the logic that links them. The two-valued
system of logic on which much of mathematics is based offers one structure
that exposes logical connections. When using this structure in
conjunction with real-world settings, which often defy the Law of the
Excluded Middle, one generally has a number of difficult decisions to
make; it is in the act of making these decisions that thoughts can become
clearer.
WATERSHED PRINCIPLE
The preservation of natural features is an issue that can be a
developer's nightmare, just as development can be the bete noir of the
environmentalist. When man-made boundaries are superimposed on the
natural environment, there is often little correspondence between the two
partitions of space. Abstractly it is not surprising, therefore, that
individuals using one way to partition space will be at loggerheads with
those using a different partition of space.
When the topography of a region is altered, it is necessarily the
case that the natural features on that surface are also altered.
Considering the contrapositive of this statement, a logical equivalent,
leads to the idea that the preservation of natural features is dependent
on the preservation of topography. When this idea is coupled with the
notion that the fundamental topographic unit is the drainage basin or
watershed (Leopold, Wolman, and Miller, Fluvial Processes in
Geomorphology), the following principle emerges.
Watershed Principle.
If the preservation of natural features depends upon the preservation
of topography and if the fundamental topographic unit is the watershed,
then the preservation of natural features depends upon the watershed.
If one accepts this Principle, then it may well be a small step to
the following Corollaries.
Corollary 1.
When environmental concerns are involved, the drainage basin should
be the fundamental planning unit.
Corollary 2.
When the drainage basin is the fundamental planning unit, the
partition of wetlands and other elements of the drainage network, by
man-made planning unit boundaries, is not possible.
Decisions as to the impact a proposed development project will have
on a wetland are facilitated by having the entire wetland contained within
the legal boundaries of the parcel; using the drainage basin as the
fundamental planning unit ensures that such set-theoretic containment will
be the case. Issues involving the welfare of the entire watershed also
become tractable under such an alignment: neighbors become neighbors with
respect to the drainage pattern rather than with respect to superimposed
human boundaries. Indeed, what my neighbor does three miles upstream from
me may have far more impact on my land that does the action of a neighbor
100 feet away who is in a different drainage basin. Current technology
(Geographic Information Systems, for example) might make it possible to
alter the inventory of lands to create suitable, substantial changes,
along these or along other lines, in legal definitions. The use of
technological capability to make legal definitions correspond more closely
to natural definitions can lead to the resolution of conflicts: the
closer the fit between natural and man-made boundaries the fewer the
disagreements.
MINIMAX PRINCIPLE
The basic idea behind the Watershed Principle might be captured as
one that minimizes damage to the environment and maximizes satisfaction of
human needs and desires. Viewed more broadly, the Watershed Principle
might be recast as a MiniMax Principle which can then be recast downstream
abstractly, in a number of other more specific forms (such as the
Watershed Principle).
MiniMax Principle
An optimal plan is one which minimizes alteration of existing
entities and maximizes the common good.
Highly general principles, such as this one, demand attention to
definitional matters: what is meant by "common good" or how might one
measure "alteration." These are difficult problems: one advantage to an
abstract view is to bring important and difficult issues into focus.
EARTH-SUN RELATIONS: GEOGRAPHIC COORDINATES AND TIME ZONES.
One case in which the fit between natural and man-made boundaries is
done in a style consistent with the Minimax Principle is the spatial
layout of reckoning time (thus, time becomes transformed in a "meta"
fashion into space). Much of the developed world measures the passage of
time by the position of Earth relative to our Sun. One unit of time, the
year, corresponds roughly to one revolution of the Earth around the Sun.
Another, smaller, unit of time, the day, corresponds roughly to the
rotation of the Earth on its axis--the man-made boundaries in both cases
are set by the natural planetary motions in space.
When planetary motions do not permit any further refinement of the
day into even smaller units, we subdivide the day into hours. When the
partition of the day into 24 hours is put into correspondence with the
grid system based on latitude and longitude, one hour corresponds to
fifteen degrees of longitude. Fifteen degrees of longitude corresponds to
a central angle of fifteen degrees intercepted along the Equatorial great
circle. Thus, 24 man-made time zones of 15 degrees of longitude each
envelop the Earth--man-made boundaries again follow (although a bit
indirectly) from natural boundaries. The Earth becomes a "clockwork
orange" of 24 sections, each 1 hour wide, with boundaries along meridians
spaced 15 degrees apart. Across oceans, this alignment of time-zone and
longitude may reasonably have boundaries along meridians; interior to a
continent, however, human needs and desires may reasonably prevail, making
it prudent to bend the natural alignment for the common good.
** The author wishes to thank her colleagues on the City of Ann Arbor
Planning Commission and in the Planning Department of the City of Ann
Arbor. The challenge and stimulation fostered by this lively Commission
helped to generate this viewpoint.
5. MAPBANK: AN ATLAS OF ON-LINE BASE MAPS
SANDRA L. ARLINGHAUS
The enclosed images contain a number of projections of the
world made from Geographic Information System (GIS) technology.
Right-click on the image (on a PC) to save it to your local hard drive.
Then open the .jpg file using Adobe PhotoShop or other software and, if
you wish, save it in a different format (such as bitmap).
When these downloaded maps are put into Windows Paintbrush (or other
software) as bitmaps and are projected from the computer screen onto the
wall (using
a data-show and overhead projector, or some such) their resolution is of
about the same quality as that of the original on-screen display in the
GIS. Thus, wall-maps can be carried around on diskette. This strategy
offers an easy way for university professors and pre-collegiate teachers
alike to give lectures with maps tailored to their needs--from base maps
for simple place-name recognition, to maps showing voting patterns by
party in presidential elections, to maps showing global vehicle
registration patterns, to detailed topographic maps. Naturally, the first
in the MapBank series of maps offered for this style of communication are
base maps.
Behrmann Equal Area Cylindrical Projection
Mercator (conformal) Projection
Robinson (compromise) Projection
Miller Cylindrical Projection
Latitude/longitude display
Mercator, with cylindrical nature evident.
Eckert IV, Equal Area
Eckert VI, Equal Area
Mollweide, Equal Area
Sinusoidal, Equal Area
Tobler's Hyper-Elliptical
6. INTERNATIONAL SOCIETY OF SPATIAL SCIENCES
July 18, 1995, the International Society of Spatial Sciences
(ISSS--I-triple-S) was founded as a division of the non-profit Community
Systems Foundation of Ann Arbor, MI. This primarily electronic society
has as board members:
Sandra L. Arlinghaus (founder), W. C. Arlinghaus, M. L. Bird, B. R.
Burkhalter, W. D. Drake, F. L. Goodman, F. Harary, J. A. Licate, A. L.
Loeb, K. E. Longstreth, J. D. Nystuen, W. R. Tobler. To follow its
activities, browse the under-construction WebSite
http://www-personal.umich.edu/~sarhaus/isss
with direct links to material of the American Geographical Society
and the Thunen Society.
internet: sarhaus@umich.edu
The focus of this new society is to place in a core position those
sciences, of spatial character, that are often relegated to the periphery
within academic institutional structure. Such sciences are, to name only
a few, geology, geography, and astronomy. In moving along this continuum
from the center of the Earth to the outer reaches of the universe, one
might imagine a whole host of sciences that could also be included (from
oceanography to atmospheric science to regional science). Thus, ISSS
offers a platform from which individuals, institutions, and professional
societies devoted to some aspect of spatial science might further their
interests.
The previous article announces one of the projects of ISSS. For the
past two years, during the developmental stages of ISSS, a continuing
project has been the development of a MapBank. This is a bank composed of
maps made by students; most of the maps are thematic maps made to
supplement student term papers or as maps to be used in the classroom by
students of Education. For teachers to use the MapBank, free of charge,
they must make a deposit of an electronic map. Currently the MapBank
numbers more than 100 electronic maps. Look for thematic maps to appear
in future issues of Solstice and on the WebSite of ISSS. There are
currently ten base maps of the world on the ISSS MapBank WebSite:
http//www-personal.umich.edu/~sarhaus/isss.
3. INDEX TO VOLUMES I (1990) TO VOL. V
Volume V, No. 2, Winter, 1994.
Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary: The Paris Metro: Is its Graph Planar?Planar graphs; The Paris Metro; Planarity and the Metro; Significance of lack of planarity. Sandra Lach Arlinghaus: Interruption! Classical interruption in mapping; Abstract variants on interruption and mapping; The utility of considering various mapping surfaces--GIS; Future directions. Reprint of Michael F. Dacey: Imperfections in the Uniform Plane. Forewords by John D. Nystuen. Original (1964) Nystuen Foreword; Current (1994) Nystuen Foreword; The Christaller spatial model; A model of the imperfect plane; The disturbance effect; Uniform random disturbance; Definition of the basic model; Point to point order distances; Locus to point order distances; Summary description of pattern; Comparison of map pattern; Theoretical model; Point to point order distances; Locus to point order distances; Summary description of pattern; Comparison of map pattern; Th eoretical order distances; Analysis of the pattern of urban places in Iowa; Almost periodic disturbance model; Lattice parameters; Disturbance variables; Scale variables; Comparison of M(2) and Iowa; Evaluation; Tables. Sandra L. Arlinghaus: Construction Zone: The Brakenridge-MacLaurin Construction. William D. Drake: Population Environment Dynamics: Course and Monograph--descriptive material.Volume V, No. 1, Summer, 1994.
Virginia Ainslie and Jack Licate: Getting Infrastructure Built. Cleveland infrastructure team shares secrets of success; What difference has the partnership approach made; How process affects products--moving projects faster means getting more public investment; difference has the partnership approach made; How process affects products--moving projects faster means getting more public investment; How can local communities translate these successes to their own settings?
Frank E. Barmore: Center Here; Center There; Center, Center Everywhere. Abstract; Introduction; Definition of geographic center; Geographic center of a curved surface; Geographic center of Wisconsin; Geographic center of the conterminous U.S.; Geographic center of the U.S.; Summary and recommendations; Appendix A: Calcula tion of Wisconsin's geographic center; Appendix B: Calculation of the geographical center of the conterminous U.S.; References. Barton R. Burkhalter: Equal-Area Venn Diagrams of Two Circles: Their Use with Real-World Data General problem; Definition of the two-circle problem; Analytic strategy; Derivation of B% and AB% as a function of r(B) and d(AB). Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, John D. Nystuen. Los Angeles, 1994 -- A Spatial Scientific Study. Los Angeles, 1994; Policy implications; References; Tables and complicated figures.
Volume IV, No. 2, Winter, 1993.
William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein, R. Tilden. Villages in Transition: Elevated Risk of Micronutrient Deficiency. Abstract; Moving from traditional to modern village life: risks during transtion; Testing for elevated risks in transition villages; Testing for risk overlap within the health sector; Conclusions and policy implicationsVolume IV, No. 1, Summer, 1993.
Sandra L. Arlinghaus and Richard H. Zander: Electronic Journals: Observations Based on Actual Trials, 1987-Present. Abstract; Content issues; Production issues; Archival issues; References
John D. Nystuen: Wilderness As Place. Visual paradoxes; Wilderness defined; Conflict or synthesis; Wilderness as place; Suggested readings; Sources; Visual illusion authors. Frank E. Barmore: The Earth Isn't Flat. And It Isn't Round Either: Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape. Abstract; Introduction; The Qibla problem; The geographic center; The center of population; Appendix; References. Sandra L. Arlinghaus: Micro-cell Hex-nets? Introduction; Lattices: Microcell hex-nets; References Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary: Sum Graphs and Geographic Information. Abstract; Sum graphs; Sum graph unification: construction; Cartographic application of sum graph unification; Sum graph unification: theory; Logarithmic sum graphs; Reversed sum graphs; Augmented reversed logarithmic sum graphs; Cartographic application of ARL sum graphs; Summary.
Volume III, No. 2, Winter, 1992.
Frank Harary: What Are Mathematical Models and What Should They Be? What are they? Two worlds: abstract and empirical; Two worlds: two levels; Two levels: derivation and selection; Research schema; Sketches of discovery; What should they be? Frank E. Barmore: Where Are We? Comments on the Concept of Center of Population. Introduction; Preliminary remarks; Census Bureau center of population formulae; Census Bureau center of population description; Agreement between description and formulae; Proposed definition of the center of population; Summary; Appendix A; Appendix B ; References. Sandra L. Arlinghaus and John D. Nystuen: The Pelt of the Earth: An Essay on Reactive Diffusion. Pattern formation: global views; Pattern formation: local views; References cited; Literature of apparent related interest.Volume III, No. 1, Summer, 1992.
Harry L. Stern: Computing Areas of Regions with Discretely Defined Boundaries. Introduction; General formulation; The plane; The sphere; Numerical examples and remarks; Appendix--Fortran program. Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg: The Quadratic World of Kinematic Waves.
Volume II, No. 2, Winter, 1991.
Reprint of Saunders Mac Lane: Proof, Truth, and Confusion, The Nora and Edward Ryerson Lecture at The University of Chicago in 1982. The fit of ideas; Truth and proof; Ideas and theorems; Sets and functions; Confusion via surveys; Cost-benefit and regression; Projection, extrapolation, and risk; Fuzzy sets and fuzzy thoughts; Compromise is confusing. Robert F. Austin: Digital Maps and Data Bases: Aesthetics versus accuracy. Introduction; Basic issues; Map production; Digital maps; Computerized data bases; User community.Volume II, No. 1, Summer, 1991.
Sandra L. Arlinghaus, David Barr, John D. Nystuen: The Spatial Shadow: Light and Dark -- Whole and Part. This account of some of the projects of sculptor David Barr attempts to place them in a formal systematic, spatial setting based on the postulates of the science of space of William Kingdon Clifford (reprinted in Solstice, Vol. I, No. 1.). Sandra L. Arlinghaus: Construction Zone--The Logistic Curve. Educational feature--Lectures on Spatial Theory.
Volume I, No. 2, Winter, 1990.
John D. Nystuen: A City of Strangers: Spatial Aspects of Alienation in the Detroit Metropolitan Region. This paper examines the urban shift from "people space" to "machine space" (see R. Horvath, Geographical Review, April, 1974) in the Detroit metropolitan regions of 1974. As with Clifford's Postulates, reprinted in the last issue of Solstice, note the timely quality of many of the observations. Sandra Lach Arlinghaus: Scale and Dimension: Their Logical Harmony. Linkage between scale and dimension is made using the Fallacy of Division and the Fallacy of Composition in a fractal setting. Sandra Lach Arlinghaus: Parallels Between Parallels. The earth's sun introduces a symmetry in the perception of its trajectory in the sky that naturally partitions the earth's surface into zones of affine and hyperbolic geometry. The affine zones, with single geometric parallels, are located north and south of the geographic parallels. The hyperbolic zone, with multiple geometric parallels, is located between the geographic tropical parallels. Evidence of this geometric partition is suggested in the geographic environment--in the d esign of houses and of gameboards.Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen: The Hedetniemi Matrix Sum: A Real-world Application. In a recent paper, we presented an algorithm for finding the shortest distance between any two nodes in a network of n nodes when given only distances between adjacent nodes (Arlinghaus, Arlinghaus, Nystuen, Geographical Analysis, 1990). In that previous research, we applied the algorithm to the generalized road network graph surrounding San Francisco Bay. Here, we examine consequent changes in matrix entries when the underlying adjacency pattern of the road network was altered b y the 1989 earthquake that closed the San Francisco--Oakland Bay Bridge.
Sandra Lach Arlinghaus: Fractal Geometry of Infinite Pixel Sequences: "Super-definition" Resolution? Comparison of space-filling qualities of square and hexagonal pixels. Sandra Lach Arlinghaus: Construction Zone--Feigenbaum's number; a triangular coordinatiztion of the Euclidean plane; A three-axis coordinatization of the plane.Volume I, No. 1, Summer, 1990.
Reprint of William Kingdon Clifford: Postulates of the Science of Space. This reprint of a portion of Clifford's lectures to the Royal Institution in the 1870s suggests many geographic topics of concern in the last half of the twentieth century. Look for connections to boundary issues, to scale problems, to self-similarity an d fractals, and to non-Euclidean geometries (from those based on denial of Euclid's parallel postulate to those based on a sort of mechanical `polishing'). What else did, or might, this classic essay foreshadow? Sandra Lach Arlinghaus: Beyond the Fractal. The fractal notion of self-similarity is useful for characterizing change in scale; the reason fractals are effective in the geometry of central place theory is because that geometry is hierarchical in nature. Thus, a natural place to look for other conn ections of this sort is to other geographical concepts that are also hierarchical. Within this fractal context, this article examines the case of spatial diffusion. When the idea of diffusion is extended to see "adopters" of an innovation as "attractors" of new adopters, a Julia set is introduced as a possible axis against which to measure one class of geographic phenomena. Beyond the fractal context, fractal concepts, such as "compression" and "space-filling" are considered in a broader graph-theoretic setting. William C. Arlinghaus: Groups, Graphs, and God. Sandra L. Arlinghaus: Theorem Museum--Desargues's Two Triangle Theorem from projective geometry.Construction Zone--centrally symmetric hexagons.
MONOGRAPH SERIES
INSTITUTE OF MATHEMATICAL GEOGRAPHY
Hard copy of Solstice is reprinted annually as a single volume in this series.
Electronic copy of Solstice, as well as of selected other monographs, is available on the web page of the Institute of Mathematical Geography (current electronic address available in various search engines).
OTHER PUBLICATIONS, PRODUCED ON-DEMAND.
Philbrick, Allen K. This Human World. Reprint.
Kolars, John F. and Nystuen, John D. Human Geography. Reprint.
Nystuen, John D. et al. Michigan Inter-University Community of Mathematical Geographers. Complete Papers. Reprint.
Griffith, Daniel A. Discussion Paper #1. Spatial Regression Analysis on the PC. 84pp.